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Let (X, d, μ) be a metric measure space satisfying the doubling condition and an L2-Poincaré inequality. Consider the nonnegative operator L generalized by a Dirichlet form on X. We will show that a solution u to (− ∂ t 2 + L) u = 0 on X × ℝ+ satisfies an α-Carleson condition if and only if u can be represented as the Poisson integral of the operator L with the trace in the generalized Morrey space L2,α(X), where α is a nonnegative function defined on a class of balls in X. This result extends the analogous characterization founded by R. Jiang, J. Xiao, D. Yang (2016) from the classical Morrey space on Euclidean space to the generalized Morrey space on the metric measure space. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Dirichlet problem in a bounded open set Ω ⊂ R n ( n ⩾ 2 ) for a class of second-order nonlinear elliptic equations with right-hand side f in L 1 (Ω) . We study the summability of entropy and weak solutions of this problem under the stronger assumption that f G (| f |) ∈ L 1 (Ω) , where G is a nonnegative increasing continuous function on [ 0 , + ∞) . We show how the summability of the solutions depends on the function G. Our conditions on G imply that L 1 + ε (Ω) ⊂ K G ⊂ L 1 (Ω) for every ε > 0 , where K G is the set of all measurable functions v on Ω such that v G (| v |) ∈ L 1 (Ω) . [ABSTRACT FROM AUTHOR]

In this paper, we deduce a new formula that explicitly expresses the cylindrical Poisson kernel of any degree and order using cylindrical Poisson polynomials. We then demonstrate how this formula can be applied to solve the Dirichlet boundary value problems for cylindrical Laplace equations. Furthermore, we examine and discuss the behaviors of this solution approach at infinity. [ABSTRACT FROM AUTHOR]

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of 2s-stable processes and exterior data, inhomogeneity in weighted L 2 -spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit s → 1 - which allows us to recover the local theory. [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with L p boundary data, 1 ≤ p ≤ ∞ . We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonlinear elliptic equations in the Krylov type, which can be viewed as a generalization of the Hessian equation for Schouten tensor. After a conformal change, we considered the Dirichlet problem for a modified Schouten tensor in the smooth closed Riemannian manifold with smooth boundary. A unique k-admissible solution can be assured under some suitable settings. [ABSTRACT FROM AUTHOR]

We consider elliptic operators in divergence form with lower order terms of the form L u = - div (A · ∇ u + b u) - c · ∇ u - d u , in an open set Ω ⊂ R n , n ≥ 3 , with possibly infinite Lebesgue measure. We assume that the n × n matrix A is uniformly elliptic with real, merely bounded and possibly non-symmetric coefficients, and either b , c ∈ L loc n , ∞ (Ω) and d ∈ L loc n 2 , ∞ (Ω) , or | b | 2 , | c | 2 , | d | ∈ K loc (Ω) , where K loc (Ω) stands for the local Stummel–Kato class. Let K Dini (Ω) be a variant of K (Ω) satisfying a Carleson-Dini-type condition. We develop a De Giorgi/Nash/Moser theory for solutions of L u = f - div g , where f and | g | 2 ∈ K Dini (Ω) if, for q ∈ [ n , ∞) , any of the following assumptions holds: (i) | b | 2 , | d | ∈ K Dini (Ω) and either c ∈ L loc n , q (Ω) or | c | 2 ∈ K loc (Ω) ; (ii) div b + d ≤ 0 and either b + c ∈ L loc n , q (Ω) or | b + c | 2 ∈ K loc (Ω) ; (iii) - div c + d ≤ 0 and | b + c | 2 ∈ K Dini (Ω) . We also prove a Wiener-type criterion for boundary regularity. Assuming global conditions on the coefficients, we show that the variational Dirichlet problem is well-posed and, assuming - div c + d ≤ 0 , we construct the Green's function associated with L satisfying quantitative estimates. Under the additional hypothesis | b + c | 2 ∈ K ′ (Ω) , we show that it satisfies global pointwise bounds and also construct the Green's function associated with the formal adjoint operator of L. An important feature of our results is that all the estimates are scale invariant and independent of Ω , while we do not assume smallness of the norms of the coefficients or coercivity of the associated bilinear form. [ABSTRACT FROM AUTHOR]

In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems. [ABSTRACT FROM AUTHOR]

We show that, under natural assumptions, solutions of Dirichlet problems for uniformly elliptic divergence form operators can be approximated pointwise by solutions of some versions of Robin problems. The proof is based on the stochastic representation of solutions and properties of reflected diffusions corresponding to divergence form operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of a new class of nonlocal boundary value problems for the generalized Helmholtz equation in the unit ball. These problems are a generalization of the classical Dirichlet boundary value problem to the Helmholtz equation. For the considered problems, existence and uniqueness theorems are proved. Integral representation of solutions is established. Corresponding spectral questions are also investigated, namely, eigenfunctions and eigenvalues of the problem are found. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 , Ω be a strongly Lipschitz domain of R n , and p (·) : R n → (0 , 1 ] a variable exponent function satisfying the globally log-Hölder continuous condition. Assume that L Ω : = - Δ + V is a Schrödinger operator on L 2 (Ω) with the Dirichlet boundary condition, where Δ denotes the Laplace operator and the nonnegative potential V belongs to the reverse Hölder class R H q 0 (R n) for some q 0 ∈ (n / 2 , ∞ ] . In this article, the authors first introduce the variable Hardy space H L Ω p (·) (Ω) associated with L Ω on Ω , via the Lusin area function associated with L Ω , and the "geometrical" variable Hardy space H L R n , r p (·) (Ω) , via the variable Hardy space H L R n p (·) (R n) associated with the Schrödinger operator L R n : = - Δ + V on R n , and then prove that H L Ω p (·) (Ω) = H L R n , r p (·) (Ω) with equivalent quasi-norms. As an application, the authors show that, when Ω is a bounded, simply connected, and semiconvex domain of R n and the nonnegative potential V belongs to the reverse Hölder class R H q 0 (R n) for some q 0 ∈ (max { n / 2 , 2 } , ∞ ] , the operators V L Ω - 1 and ∇ 2 L Ω - 1 are bounded from H L R n , r p (·) (Ω) to the variable Lebesgue space L p (·) (Ω) , or to itself. As a corollary, the second-order regularity for the inhomogeneous Dirichlet problems of the corresponding Schrödinger equations in the scale of variable Hardy spaces H L R n , r p (·) (Ω) is obtained. [ABSTRACT FROM AUTHOR]

We discuss the solvability of nonlinear Dirichlet problems of the type - Δ p , w u = σ in Ω ; u = 0 on ∂ Ω , where Ω is a bounded domain in R n , Δ p , w is a weighted (p, w)-Laplacian and σ is a nonnegative locally finite Radon measure on Ω . We do not assume the finiteness of σ (Ω) . We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by L p (w) - L q (σ) trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem - Δ p , w u = σ u - γ in Ω ; u = 0 on ∂ Ω and derive connection with the trace inequalities. [ABSTRACT FROM AUTHOR]

We use the concept of the inside-(a, η, h) domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space. As an application, we prove that the Hölder exponent max{1/a, 1/(n + 1)} for the problem is optimal for any a ∈ [2, +∞]. [ABSTRACT FROM AUTHOR]

In the half-plane, the Dirichlet problem is considered for elliptic differential-difference equations with nonlocal general-kind potentials, which are linear combinations of translations of the desired function, not bounded by commensurability conditions. We find a condition for the symbol of the corresponding differential-difference operator, providing the classical solvability of the specified problem for each continuous and bounded boundary-value function. The representation of the specified classical solution by a Poisson-type integral is constructed. [ABSTRACT FROM AUTHOR]

We show continuity in generalized weighted Orlicz-Morrey spaces M w Φ , φ of sublinear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operators with discontinuous data. [ABSTRACT FROM AUTHOR]

In this paper we prove the nonexistence of nontrivial solutions to − Δ u = f (u) in Ω , u = 0 on ∂ Ω , with Ω ⊂ R N ( N ≥ 1) a bounded domain and f locally Lispchitz with nonpositive primitive. As a consequence, we discuss the long-time behavior of solutions to the so-called sine-Gordon equation. [ABSTRACT FROM AUTHOR]

In this paper we establish an asymptotic expansion near the boundary for solutions to the Dirichlet problem of elliptic equations with singularities near the boundary. This expansion formula shows the singularity profile of solutions at the boundary. We deal with both linear and nonlinear elliptic equations, including fully nonlinear elliptic equations and equations of Monge-Ampère type. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in R d , d ≥ 2 . A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if p ≠ 2 , the classical results indicate that the W 1 , p estimate of the solution may go to infinity as the size of the hole tends to zero. With the presence of the shrinking hole in a fixed domain, we give a complete description for the uniform W 1 , p estimates of the solution for all 1 < p < ∞ . We show that the uniform W 1 , p estimate holds if and only if d ′ < p < d ( p = 2 when d = 2 ). We then give two applications in the study of homogenization problems in fluid mechanics: a generalization of the restriction operator and a construction of Bogovskii type operator in perforated domains with a quantitative estimate of the operator norm. [ABSTRACT FROM AUTHOR]

Foundations of the modified Poisson kernel method were laid out by Finkelstein and Scheinberg in 1975 in the context of explicit solvability for the adaptive Dirichlet problem in a half plane. Over the past decade, this method has been further developed, and new applications have appeared both in the field of harmonic analysis and operator theory and in practical Schrödinger problems. In this paper, we pay special attention to cylindrical Poisson kernel method's application for the integral representations of harmonic functions in a cylinder. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Dirichlet problem for harmonic functions on a two-dimensional complex of a special type. We prove that this problem is a Fredholm problem in the Hölder class and its index is zero. [ABSTRACT FROM AUTHOR]

We prove second-order a priori estimate on the boundary for the Dirichlet problem of a class of fully nonlinear equations on compact almost Hermitian manifolds with smooth boundary. As applications, we solve the Dirichlet problem of the Monge–Ampère type equation and of the degenerate Monge–Ampère equation. [ABSTRACT FROM AUTHOR]

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular. [ABSTRACT FROM AUTHOR]

We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak- A ∞ property) of caloric measure with respect to surface measure, for an open set Ω ⊂ ℝn+ 1, assuming as a background hypothesis only that the essential boundary of Ω satisfies an appropriate parabolic version of Ahlfors-David regularity, entailing some backwards in time thickness. Since the weak- A ∞ property of the caloric measure is equivalent to Lp solvability of the initial-Dirichlet problem, we may then deduce that BMO-solvability implies Lp solvability for some finite p. [ABSTRACT FROM AUTHOR]

For each pair ε = (ε 1 , ε 2) of positive parameters, we define a perforated domain Ω ε by making a small hole of size ε 1 ε 2 in an open regular subset Ω of ℝ n ( n ≥ 3 ). The hole is situated at distance ε 1 from the outer boundary ∂ Ω of the domain. Thus, when ε → (0 , 0) both the size of the hole and its distance from ∂ Ω tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain Ω ε and we denote its solution by u ε. Our aim is to represent the map that takes ε to u ε in terms of real analytic functions of ε defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of u ε to suitable subsets of Ω ε , we prove a global representation formula that holds on the whole of Ω ε. Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct. [ABSTRACT FROM AUTHOR]

The goal of this paper is to investigate a parametric Dirichlet problem with (p , q) -Laplacian and concave–convex nonlinearity. Denoting by S λ the set of positive solutions of the problem corresponding to the parameter λ > 0 , we prove that the set S λ is compact in the W 0 1 , p (Ω) -topolgy. We also establish an upper semicontinuity and a Mosco convergence result for the multivalued mapping λ ↦ S λ . [ABSTRACT FROM AUTHOR]

A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix different from the one widely used in the literature by the authors Chkadua, Mikhailov and Natroshvili. Mapping properties of the surface and volume parametrix-based potential-type operators are analysed. Invertibility of the single layer potential is also studied in detail in appropriate Sobolev spaces. We show that the system of boundary-domain integral equations derived is equivalent to the Dirichlet problem prescribed and we prove the existence and uniqueness of solution in suitable Sobolev spaces of the system obtained by using arguments of compactness and Fredholm Alternative theory. A discussion of the possible applications of this new parametrix is included. [ABSTRACT FROM AUTHOR]

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- A ∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ⊂ R n + 1 with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω , with data in L p (∂ Ω) for some p < ∞ . In this paper, we give a geometric characterization of the weak- A ∞ property, of harmonic measure, and hence of solvability of the L p Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp. [ABSTRACT FROM AUTHOR]

We establish gradient estimates for solutions to the Dirichlet problem for the constant mean curvature equation in hyperbolic space. We obtain these estimates on bounded strictly convex domains by using the maximum principles theory of Φ-functions of Payne and Philippin. These estimates are then employed to solve the Dirichlet problem when the mean curvature H satisfies H < 1 under suitable boundary conditions. [ABSTRACT FROM AUTHOR]

Second-order singular elliptic equations of the form ∑ i = 1 n ∂ 2 u ∂ x i 2 + γ i x i ∂ u ∂ x i = b 2 u are considered. For a b>0 we find an analytical solution to the Dirichlet problem for this equation. A procedure for obtaining its solution is proposed based on the using of a modified Bessel function of the second kind. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for a second order linear elliptic partial differential equation with discontinuous coefficients in unbounded domains. We establish an existence and uniqueness result. [ABSTRACT FROM AUTHOR]

The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is considered. The existence of an entropy solution in anisotropic Sobolev spaces with variable exponents is established. It is proved that the obtained entropy solution is a renormalized solution of the considered problem. [ABSTRACT FROM AUTHOR]

In the present article we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with L1-right-hand sides in a bounded domain of ℝn(n ⩾ 2) . This class is described by the presence of a set of exponents q1,…, qn and a set of weighted functions ν1,…, νn in growth and coercitivity conditions on coefficients of the equations. The exponents qi characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions νi characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to investigate the existence of entropy solutions of the problem under consideration.

We construct a bounded C1 domain Ω in Rn for which the H3/2 regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in C∞(Ω¯) such that the solution of Δu=f in Ω and either u=0 on ∂Ω or ∂nu=0 on ∂Ω is contained in H3/2(Ω) but not in H3/2+ε(Ω) for any ε>0. An analogous result holds for Lp Sobolev spaces with p∈(1,∞). [ABSTRACT FROM AUTHOR]

We consider the Dirichlet problem in an arbitrary unbounded domain with inhomogeneous boundary conditions for a certain class of anisotropic elliptic equations whose right-hand sides belong to the class L1 and prove the existence of entropic solutions in anisotropic Sobolev–Orlicz spaces. [ABSTRACT FROM AUTHOR]

We consider a nonlinear nonparametric elliptic Dirichlet problem driven by the p-Laplacian and reaction containing a singular term and a (p - 1) -superlinear perturbation. Using variational tools together with suitable truncation and comparison techniques we produce two positive, smooth, ordered solutions. [ABSTRACT FROM AUTHOR]

We investigate the concavity region of solutions to the Poisson equation on planar-bounded domains, by characterizing the nodal set D of the Hessian determinant of the solutions. In the case of solutions to the homogeneous Dirichlet problem on convex polygons, we prove that D is a smooth curve that touches every side of the polygon in exactly one point. [ABSTRACT FROM AUTHOR]

Using an improved version of the infinite dimensional theorem on diffeomorphisms, we obtain conditions for the existence of the unique weak solution of Dirichlet problem, where is a bounded domain with a -boundary and is a -Carathéodory function. It is also obtained that the solution operator, which assigns to each the unique solution of the equation, is a diffeomorphism. [ABSTRACT FROM AUTHOR]

The classical result of Ambrosetti and Prodi [1], in the form of Berger and Podolak [3], gives the exact number of solutions for the problem depending on the real parameter , for a class of convex g(u). Here, (where is the principal eigenfunction of the Laplacian on D, and is a smooth domain). By considering generalized harmonics, we give a similar result for the problem with. Such problems occur, for example, in 'fishing' applications that we discuss, and propose a new model with sign-changing solutions. Our approach also produces a very simple proof of the anti-maximum principle of Clément and Peletier [4]. [ABSTRACT FROM AUTHOR]

We combine the calculus of conormal distributions, in particular the Pull‐Back and Push‐Forward Theorems, with the method of layer potentials to solve the Dirichlet and Neumann problems on half‐spaces. We obtain full asymptotic expansions for the solutions, show that boundary layer potential operators are elements of the full b‐calculus and give a new proof of the classical jump relations. En route, we improve Siegel and Talvila's growth estimates for the modified layer potentials in the case of polyhomogeneous boundary data. The techniques we use here can be generalised to geometrically more complex settings, as for instance the exterior domain of touching domains or domains with fibred cusps. This work is intended to be a first step in a longer program aiming at understanding the method of layer potentials in the setting of certain non‐Lipschitz singularities that can be resolved in the sense of Melrose using manifolds with corners and at applying a matching asymptotics ansatz to singular perturbations of related problems. [ABSTRACT FROM AUTHOR]

It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary of an open set Ω ⊂ ℝn+1 with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in Lp(∂ Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with contin¬uous boundary data, can be solved, one may seek to characterize the open sets for which Lp solvability holds, thus allowing for singular boundary data. It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ∂ Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric charac¬terization of the weak-A∞ property, and hence of solvability of the Lp Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa. [ABSTRACT FROM AUTHOR]

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L0 = divA0(x)∇ + B0(x). ∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the upper half-space ℝ+n. In this paper we prove that the Lp solvability is stable under small perturbations of L0. That is if L1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L0 and L1 are sufficiently close in the sense of Carleson measures, then the Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp solvability of the Dirichlet problem for operators of the form L = divA(x)∇ + B(x) · ∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiate and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindoš, Petermichl and Pipher. [ABSTRACT FROM AUTHOR]

Brosamler's formula gives a probabilistic representation of the solution of the Neumann problem for the Laplacian on a smooth bounded domain D ⊂ R n in terms of the reflecting Brownian motion in D. The original proof, as well as other proofs in the literature (e.g., in the case of Lipschitz domains), are based on potential theory (transition densities of the reflecting Brownian motion). We give new proofs of Brosamler's formula using (path trajectories of) stochastic processes. More precisely, we use a connection between the Dirichlet and the Neumann boundary problems, and the explicit description of the reflecting Brownian motion and its boundary local time in terms of the free Brownian motion. The results are obtained in the case of the Euclidean unit ball in any dimension and in the case of smooth C 1 , α planar simply connected domains, for continuous boundary data, and then extended to the case of bounded measurable data, respectively integrable boundary data. A new Brosamler-type formula in terms of the free Brownian motion is also given. [ABSTRACT FROM AUTHOR]

We consider solving the exterior Dirichlet problem for the Helmholtz equation with the h-version of the boundary element method (BEM) using the standard second-kind combined-field integral equations. We prove a new, sharp bound on how the number of GMRES iterations must grow with the wavenumber k to have the error in the iterative solution bounded independently of k as k → ∞ when the boundary of the obstacle is analytic and has strictly positive curvature. To our knowledge, this result is the first-ever sharp bound on how the number of GMRES iterations depends on the wavenumber for an integral equation used to solve a scattering problem. We also prove new bounds on how h must decrease with k to maintain k-independent quasi-optimality of the Galerkin solutions as k → ∞ when the obstacle is nontrapping. [ABSTRACT FROM AUTHOR]

We prove correctness of the Kato square root conjecture for elliptic differential‐difference operators with degeneration. [ABSTRACT FROM AUTHOR]

We show that for a uniformly elliptic divergence form operator L, defined in an open set Ω with Ahlfors-David regular boundary, BMO solvability implies scale-invariant quantitative absolute continuity (the weak-A∞ property) of elliptic-harmonic measure with respect to surface measure on ∂Ω. We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of Ω. In this generality, our results are new even for the Laplacian. Moreover, we obtain a partial converse, assuming in addition that Ω satisfies an interior Corkscrew condition, in the special case that L is the Laplacian. [ABSTRACT FROM AUTHOR]